Non-analytical image reconstruction is a generally known approach for a volumetric reconstruction, most commonly used for emission tomographic data, based e.g. on SPECT (single photon emission computed tomography) or PET (positron emission tomography) measurements (see K. Lange et al. “EM Reconstruction Algorithms for Emission and Transmission Tomography” in “Journal of Computer Assisted Tomography”, vol. 8 (2), April 1984, p. 306-316, L. A. Shepp et al. “Maximum Likelihood Reconstruction for Emission Tomography” in “IEEE Transactions on Medical Imaging” vol. MI-1 (2), October 1982, p. 113-122), and J. Kay “The EM algorithm in medical imaging” in “Statistical Methods in Medical Research” vol. 6, 1997, p. 55-75). In contrast to the analytical approach, in particular using a back-projection algorithm (e.g. filtered back-projection, FBP), the non-analytical solution uses a minimizing algorithm for finding image data which represent the best fit for the detector data. The non-analytical approach intrinsically can cope better with input data suffering from noise and is able to account for physical processes degrading the measured data. The measured data often suffer from very bad photon statistics. In order to obtain useful results, long measurement times and relatively high patient doses are necessary, putting unnecessary stress on the patient. These restrictions would be avoided if images could be reconstructed from few measurements with the concept of Compressive Sensing (CS) as proposed e.g. for conventional optical images collected with a camera.
Compressed Sensing (or: compressive sensing, compressive sampling and sparse sampling), is a technique for acquiring and reconstructing a signal utilizing the prior knowledge that it is sparse or compressible. An introduction to CS has been presented by Emmanuel J. Candès et al. (“An Introduction To Compressive Sampling” in “IEEE Signal Processing Magazine” March 2008, p. 21-30). The CS theory shows that sparse signals, which contain much less information than could maximally be encoded with the same number of detected data entries, can be reconstructed exactly from very few measurements in noise free conditions.
According to Jarvis Haupt et al. (“Signal reconstruction from noisy random projections” in “IEEE Trans. on Information Theory” vol. 52(9), 2006, p. 4036-4048), a set of data fj* of size v (j=1, . . . , v) which is sparse (or rather “compressible”) can be accurately reconstructed from a small number k of random projections even if the projections are contaminated by noise of constant variance, e.g. Gaussian noise. Specifically, yi=Σjφijfj*+ξl with i=1, . . . , k are the noisy projections of fj*, taken with the projection matrix φij which consists of random entries all drawn from the same probability distribution with zero mean and variance 1/v and the noise ξl is drawn from a Gaussian probability distribution with zero mean and variance σ2. By finding the minimizer {circumflex over (f)}j of a certain functional (to be shown in detail below), one obtains an approximation to fj* for which the average error is bounded by a constant times (k/log v)−a with 0<a≦1, i.e. the error made depends only logarithmically on v. To put it another way, the error can be made small by choosing k/log v large, but it is by no means necessary to have k/v close to 1. Accurate reconstruction is possible even if the number of projections is much smaller than v, as long as k/log v is large.
A crucial point in the derivation of the above result is the fact that the variance of the noise ξi is a constant. Even though a similar results could be obtained for non-Gaussian noises (provided certain noise properties required for the validity of the Craig-Bernstein inequality can be proved), the result does not easily carry over to the case that the variance of the noise depends on the values fj*. Yet this is precisely what happens e.g. in photon limited imaging systems or in emission tomography where a main source of noise is the discrete quantum nature of the photons. In this case the projections yi have Poisson statistics with parameter μi=Σjφijfj*. This parameter is equal to the mean of yi but also to its variance.
In the past, it was tested whether the principle of accurate reconstructions from few projections carries over to Poisson noise in order to make accurate reconstructions possible with fewer measurements, e.g. in emission tomography. It was expected that the compressive sensing strategy for the reconstruction of sparse or compressible objects from few measurements is difficult to apply to data corrupted with Poisson noise, due to the specific properties of Poisson statistics and the fact that measurements can not usually be made randomly, as in many other cases.
Rebecca M. Willett et al. (“Performance bounds on compressed sensing with Poisson noise” in “IEEE International Symposium on Information Theory ISIT” 2009, Jun. 28, 2009-Jul. 3, 2009, p. 174-178) have generalized results from Jarvis Haupt et al. to Poisson noise. In particular, Rebecca M. Willett et al. have proposed to reconstruct a tomographic image from detector data using a procedure of minimizing a functional {circumflex over (f)} depending on a sensing matrix A and the detector data and further depending on a penalty term, wherein the sensing matrix A is constructed on the basis of statistic Rademacher variables and the penalty term depends on the sparsity of the object. However, the result was discouraging: it was found that the upper bound on the error increases with the number of measurements, i.e. more measurements seem to make the accuracy smaller.
There is a strong interest in imaging techniques where the main source of noise is the Poisson statistics of the photons, e.g. in emission tomography, to apply CS strategies, in particular for obtaining shorter acquisition times and reduced motion artifacts.